Nevanlinna Theory in Several Complex Variables and Diophantine Approximation

425 Pages·2014·2.03 MB·English

by  Junjiro Noguchi

Grundlehren der
mathematischen Wissenschaften 350
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M.Berger P.delaHarpe N.J. Hitchin
A.Kupiainen G.Lebeau F.-H.Lin
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Editor-in-Chief
A.Chenciner J.Coates S.R.S.Varadhan
Forfurthervolumes:
www.springer.com/series/138
Junjiro Noguchi (cid:2) Jörg Winkelmann
Nevanlinna
Theory in Several
Complex Variables
and Diophantine
Approximation
JunjiroNoguchi(emeritus) JörgWinkelmann
TheUniversityofTokyo Ruhr-UniversityBochum
Tokyo,Japan Bochum,Germany
and
TokyoInstituteofTechnology
Tokyo,Japan
ISSN0072-7830 ISSN2196-9701(electronic)
GrundlehrendermathematischenWissenschaften
ISBN978-4-431-54570-5 ISBN978-4-431-54571-2(eBook)
DOI10.1007/978-4-431-54571-2
SpringerTokyoHeidelbergNewYorkDordrechtLondon
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Preface
BasedontheoremsofPicardandBorel,R.Nevanlinnapublishedhispaper[25]and
evolved a theory that carries his name. The development of the theory into com-
plex analysis in several variables was quite fast and a number of papers such as
A.Bloch[26b],H.Cartan[33],andH.Weyl–J.Weyl[38]werepublished.Follow-
ing H. Weyl–J. Weyl [38], the paper of A.L. Ahlfors [41] was published, and this
work was succeeded by W. Stoll [53b], [54], being developed into the theory of
meromorphicmappingsfromparabolicspacesintocomplexprojectivespaces.
S.-S.Chernstudiedthevaluedistributionofholomorphiccurvesfromtheview-
pointofdifferentialgeometry,andpublishedanumberofpapers;oneofthemwasa
joint paper, Bott–Chern [65], where they introduced a “connection” to the study
of the value distribution theory and found a relation with characteristic classes.
S.-S.Chern introduced with Osserman the Nevanlinna theoretic method into the
study of Gauss’ maps of minimal surfaces in Euclidean spaces. This approach
was advanced much, especially by H. Fujimoto, and finally reached the solution
ofGauss’mapconjecture(H.Fujimoto[88a]).
Fromthelate1960stotheearly1970sthetheoryofKobayashihyperbolicman-
ifoldsandthevaluedistributiontheoryofequidimensionalholomorphicmappings
were initiated (Kobayashi [67], [70]; Carlson–Griffiths [72]; Griffiths–King [73]),
and the value distribution theory in several complex variables advanced consider-
ably and was enlarged, so that it has broadened the border with more related sub-
jects.
T.Ochiai[77]revivedBloch’spaper(Bloch[26b])andprovedalemmaonholo-
morphicdifferentials,whichisacounterparttothelemmaonlogarithmicderivatives
inNevanlinnatheory,andobtainedtheso-calledtheBloch–OchiaiTheorem.Inthe
studyof theextensionof theBloch–OchiaiTheoremtothenon-compactcase, the
relationwithP.Deligne’slogarithmicforms(Deligne[71],[74])wasfoundandthe
initialsteptodealwithvarietiesotherthancomplexprojectivespacebeganforthe
value distribution theory of holomorphic curves (Noguchi [77b]). As a result, the
Bloch–OchiaiTheoremandtheclassicaltheoremofBorelwereunified.
Ontheotherhand,S.Langformulatedaconjectureofahigherdimensionalver-
sion of Mordell’s conjecture on rational points such that a Kobayashi hyperbolic
v
vi Preface
algebraic manifold defined over a number field would have at most finitely many
rational points (Lang’s conjecture), and proposed a number of related conjectures
to enhance the study of problems over function fields and of holomorphic curves
(Mordell [22]; Lang [74], [86]). In 1987 P. Vojta [87], completing the viewpoint
ofOsgood[81],formulatedananaloguebetweenNevanlinnatheoryandDiophan-
tineapproximationtheory.Heproposedthewell-knownVojtaconjecture,whichhas
attractedtheattentionofmanyresearchers.
As for monographs describing these developments of the study of Nevanlinna
theory and Diophantine approximation theory, we may list, in order of year, e.g.,
Kobayashi [70], Wu [70], Stoll [70], [73], Griffiths [76], Stoll [77a], Lang [83],
Shiffman[83],Ochiai–Noguchi[84],Shabat[85],Vojta[87],Lang[87],Noguchi–
Ochiai [90], Lang [91], Fujimoto [93], Kobayashi [98], Ru [01] and Bombieri–
Gubler[06].
On Kobayashi’s hyperbolicity, his book [98] provides in great detail the results
obtained up to the present. Fujimoto [93] describes the theory of Gauss’ maps of
minimalsurfacesinaself-containedmanner.
ThecoreofNevanlinnatheoryconsistsoftwoMainTheorems.TheFirstMain
Theorem is considered to be a non-compact version of Poincaré’s duality, and we
nowhaveasatisfactorytheoryforit.Inthepresentbookthesecondchapterisde-
voted to it. On the other hand, there are not many cases where the Second Main
Theorem is established for meromorphic mappings f :Cn →M and divisors D
onM;thefollowingisessentiallythefulllist:
(i) n=1, M =Pm(C), D is a union of hyperplanes in general position (Car-
tan [33]; H. Weyl–J. Weyl [38]; Ahlfors [41]), Nochka’s solution of Cartan’s
conjecture(Nochka[83];Chen[90]).
(ii) n(cid:2)1, M =Pm(C), D is a union of hyperplanes in general position (Stoll
[53b],[54];Chen[90]).
(iii) n = 1, M = Pm(C), D is a union of hypersurfaces in general position
(Eremenko–Sodin[92];Corvaja–Zannier[04b];Ru[04]).
(iv) n(cid:2)dimM and f is differentiably non-degenerate (of maximal rank at some
point) (Griffiths’ theory, Carlson–Griffiths [72]; Griffiths–King [73]; Sakai
[74a];Shiffman[75];Noguchi[76a]).
(v) M is an abelian or semi-abelian variety and D is an arbitrary divisor on it.
WhenM isasemi-abelianvariety,acompactificationofM istakensothatthe
¯
closureD isingeneralposition(inasense)(Noguchi–Winkelmann–Yamanoi
[00],[02],[08]).
(vi) M=P1(C),n=1andDisamovingdivisorforwhichthecountingfunctions
aretruncatedtolevelone(K.Yamanoi[04d],[05],[06]).
Thereisnobookamongthelistabovementionedthatdealswithalloftheabove
cases. Because of the applications to the degeneracy problem and Kobayashi hy-
perbolicity, we deal here with the cases of (i), (ii), (iv), and (v), and describe the
theory of the all Second Main Theorems in these cases from a uniformized view-
point in a self-contained way (Chaps. 3, 4 and 6). For the other cases, we limit
ourselvesonlytointroducingsomeimportantresults.TheKobayashihyperbolicity
Preface vii
willbediscussedinChap.7,andwewilldescribesomenewresultswhichwerenot
mentionedinKobayashi[98].Thesearethefirstpurposeofthisbook.
ThesecondpurposeistodescribeanddiscusstheanalogybetweenNevanlinna
theory and Diophantine approximation theory. The case of function fields is dealt
withinChap.8,andthecaseofnumberfieldsinChap.9.
Thereaderswehaveinmindaregraduatestudentsandresearcherswhoareinter-
estedinthesesubjects.Foraneasierunderstandingoftheideaofthedevelopmentof
thetheorywestartwithNevanlinnatheoryformeromorphicfunctionsontheone-
dimensionalcomplexplane.Asprerequisitesthebasictheoryofanalyticfunctions
in one variable, the very elementary part of analytic functions in several complex
variables, and the terminologies of the theory of complex manifolds are assumed.
Othermaterialswhichweuseinthetextwithoutgivingtheproofswillbeexplained
atleastsothatthereadersareabletounderstandthecontents.
Wewillbrieflymentionthecontentofeachchapterinthesequel.
InChap.1weprovetheFirstandtheSecondMainTheoremsinaself-contained
mannerformeromorphicfunctionsontheGaussianplane,whichisthemostbasic
partofNevanlinnatheory.Thisprovidestheprototypeofthetheoryextendedtothe
higherdimensionalcase.
In Chap. 2 plurisubharmonic functions are described and Jensen’s formula is
proved.ThenweshowtheFirstMainTheoremformeromorphicmappingsfromthe
m-dimensional complex space Cm into a compact complex manifold with respect
to holomorphic line bundles and coherent ideal sheaves of the structure sheaf. We
will introduce a various fundamentalproperties of order functions. The content of
this chapter serves the basic for the research of this subject, and we intended that
thereadersnaturallymasterthefundamentalmethodandknowledgeofthissubject.
In Chap. 3 we first extend the lemma on logarithmic derivatives to the case of
several complex variables. We will give a simplified proof for it. Then we prove
theSecondMainTheorembyusingthelemmaonlogarithmicderivatives,whichis
differenttothemetricmethodduetoGriffithsetal.Wealsogivesomeapplications.
In Chap. 4 we deal with the theory of holomorphic curves from the complex
plane,whichwecall“entirecurves”.Wedescribe,followingtoChen[90],theno-
tionoftheNochkaweights,whichplaysanimportantroleintheproofofCartan’s
conjecture,nowNochka’stheorem.Lemmaonlogarithmicjetdifferentialsisproved
in a most generalized and uniformized form due to Noguchi [77b], [81b], [85b],
[86],andVitter[77].UsingthisweproveaninequalityoftheSecondMainTheo-
remtypeforentirecurvesinanalgebraicvariety,andthendeducetheLogarithmic
Bloch–OchiaiTheorem(Noguchi[77b],[81b]).
Chapter 5 is devoted to the semi-tori and their compactifications. The contents
arethepreparationforthenextchapter.
InChap.6wethenestablishthelateresultoftheSecondMainTheoremforentire
curvesinsemi-abelianvarieties(Noguchi–Winkelmann–Yamanoi[00],[02],[08]).
From this Lang’s conjecture for entire curves is derived, and more applications
are given (Yamanoi [04b]; Noguchi–Winkelmann–Yamanoi [07], [13]; Winkel-
mann[11];Corvaja–Noguchi[12];Lu–Winkelmann[12]).
viii Preface
InChap.7wedescribetheapplicationsoftheresultsofChaps.4and6totheal-
gebraicdegeneracyproblemofentirecurvesandtheKobayashihyperbolicityprob-
lem.Inparticular,beingrelatedtotheKobayashiconjecture,theexistenceofanal-
gebraic curve of degree 5 in P2(C) whose complement is hyperbolic and hyper-
bolically embedded into P2(C) is proved (Zaidenberg [89]). We construct explicit
examples of hyperbolic hypersurfaces of Pn(C) for arbitrary n, and those whose
complementarehyperbolicandhyperbolicallyembeddedintoPn(C).
InChap.8wedescribeNevanlinnatheoryoverfunctionfields.Thisismotivated
bytheconjecturesofS.LangandP.Vojta.Nevanlinnatheoryoverfunctionfieldsis
understood as an approximationtheory of rational functions by rational functions.
Weproveananalogueoftheso-calledabc-conjectureofMasserandOesterléover
functionfields(cf.Oesterlé[88],Sect.3;Granville–Tucker[02]).
InChap.9wedescribeDiophantineapproximationtheoryovernumberfieldsin
an analogous way to Nevanlinna theory, based on Vojta’s idea. As application we
provefinitenesstheoremsofintegralpoints.Readerswillfindanalogiesnotonlyin
conjecturesorresultsbutalsoeveninproofs.
We would like to draw attention to Nevanlinna theory of p-adic analytic func-
tions, started by Ha (Ha Hui Khoai [83]; Ha Hui Khoai–Tu [95]), which is not
mentionedinthisbook.
ThereisalargeundevelopedareaintherelationbetweenNevanlinnatheoryof
entire curves and Diophantine approximation theory, which is expected to be ex-
ploredmore.Wehopethatthepresentbookwillserveforreadersanintroductionto
thissubject.
In writing this book the authors received a valuable comment on Nochka’s
weights from Professor N. Toda, and helpful advice on Chap. 9 from Professor
N.Hirata.ProfessorsY.AiharaandK.Yamanoihavegivenanumberofinvaluable
comments all through the book. The authors express their sincere gratitude to all
ofthem.Takingthisopportunity,theauthorsthankProfessorsMitsuruOzawa,Wil-
helm Stoll, Shoshichi Kobayashi, Hirotaka Fujimoto, Takushiro Ochiai, and Alan
Huckleberryfortheirteachings,varioussuggestionsandcommentsallthroughtheir
researchlives.
They also deeply thank the anonymous reviewers for their many helpful com-
mentsandsuggestions,inparticular,onewhoreadChap.9verycarefullyandsug-
gestedanumberofvaluablecomments.
Tokyo JunjiroNoguchi
Bochum JörgWinkelmann
February2013
Contents
1 NevanlinnaTheoryofMeromorphicFunctions . . . . . . . . . . . 1
1.1 TheFirstMainTheorem . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheSecondMainTheorem . . . . . . . . . . . . . . . . . . . . 12
1.3 ExamplesofFunctionsofFiniteOrder . . . . . . . . . . . . . . 19
2 TheFirstMainTheorem . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 PlurisubharmonicFunctions . . . . . . . . . . . . . . . . . . . . 25
2.1.1 OneVariable . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 SeveralVariables. . . . . . . . . . . . . . . . . . . . . . 33
2.2 Poincaré–LelongFormula . . . . . . . . . . . . . . . . . . . . . 42
2.3 TheFirstMainTheorem . . . . . . . . . . . . . . . . . . . . . . 50
2.3.1 MeromorphicMappings,DivisorsandLineBundles . . . 50
2.3.2 DifferentiableFunctionsonComplexSpaces . . . . . . . 54
2.3.3 MetricsandCurvatureFormsofLineBundles . . . . . . 58
2.4 TheFirstMainTheoremforCoherentIdealSheaves . . . . . . . 66
2.4.1 ProximityFunctionsforCoherentIdealSheaves . . . . . 66
2.4.2 TheCaseofm=1 . . . . . . . . . . . . . . . . . . . . . 71
2.5 OrderFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.2 Cartan’sOrderFunction . . . . . . . . . . . . . . . . . . 77
2.5.3 AFamilyofRationalFunctions . . . . . . . . . . . . . . 79
2.5.4 CharacterizationofRationality . . . . . . . . . . . . . . 83
2.6 Nevanlinna’sInequality . . . . . . . . . . . . . . . . . . . . . . 86
2.7 RamifiedCoversoverCm . . . . . . . . . . . . . . . . . . . . . 88
3 DifferentiablyNon-degenerateMeromorphicMaps . . . . . . . . . 91
3.1 LemmaonLogarithmicDerivatives . . . . . . . . . . . . . . . . 91
3.2 TheSecondMainTheorem . . . . . . . . . . . . . . . . . . . . 93
3.3 ApplicationsandGeneralizations . . . . . . . . . . . . . . . . . 102
3.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.2 Non-KählerCounter-Example . . . . . . . . . . . . . . . 105
3.3.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . 110
ix